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A stochastic differential equation is nothing more than a short-hand notation for a corresponding integral equation. So the initial SDE you provided actually means $$ \int_0^t d S_u = \int_0^t \mu(S_u, u) du + \int_0^t\sigma(S_u, u) dW_u$$ This is how the SDE is defined (see e.g. here). The reason is that you cannot differentiate a Brownian motion. It does ...


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@Olaf gave a clear answer. Another way to see this is as system of 2 SDEs: \begin{cases} dS_u &= \mu(S_u,u) du + \sigma(S_u,u) dW_u \\ dy_u &= S_u du + S_u dW_u. \end{cases} E.g. if we want to simulate this system using Euler discretuzation then we perform \begin{cases} S_u + \Delta S_u &= S_u + \mu(S_u,u) \Delta t + \sigma(S_u,u) \epsilon ...



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